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Bounded lattices with antitone involutions and properties of MV-algebras

100%
Chajda I., Emanovský P.
Discussiones Mathematicae - General Algebra and Applications
|
2004
|
tom 24
|
nr 1
31-42
EN
We introduce a bounded lattice L = (L;∧,∨,0,1), where for each p ∈ L there exists an antitone involution on the interval [p,1]. We show that there exists a binary operation · on L such that L is term equivalent to an algebra A(L) = (L;·,0) (the assigned algebra to L) and we characterize A(L) by simple axioms similar to that of Abbott's implication algebra. We define new operations ⊕ and ¬ on A(L) which satisfy some of the axioms of MV-algebra. Finally we show what properties must be satisfied by L or A(L) to obtain all axioms of MV-algebra.
2
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Minimal bounded lattices with an antitone involution the complemented elements of which do not form a sublattice

100%
Chajda I., Länger H.
Discussiones Mathematicae - General Algebra and Applications
|
2008
|
tom 28
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nr 2
251-259
EN
Bounded lattices with an antitone involution the complemented elements of which do not form a sublattice must contain two complemented elements such that not both their join and their meet are complemented. We distinguish (up to symmetry) eight cases and in each of these cases we present such a lattice of minimal cardinality.
3
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A common approach to directoids with an antitone involution and D-quasirings

88%
Chajda I., Kolařík M.
Discussiones Mathematicae - General Algebra and Applications
|
2008
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tom 28
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nr 2
139-145
EN
We introduce the so-called DN-algebra whose axiomatic system is a common axiomatization of directoids with an antitone involution and the so-called D-quasiring. It generalizes the concept of Newman algebras (introduced by H. Dobbertin) for a common axiomatization of Boolean algebras and Boolean rings.
4
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Congruences on semilattices with section antitone involutions

75%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2010
|
tom 30
|
nr 2
207-215
EN
We deal with congruences on semilattices with section antitone involution which rise e.g., as implication reducts of Boolean algebras, MV-algebras or basic algebras and which are included among implication algebras, orthoimplication algebras etc. We characterize congruences by their kernels which coincide with semilattice filters satisfying certain natural conditions. We prove that these algebras are congruence distributive and 3-permutable.
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